A Dynamic programming problem

Question

Can anyone help me find an optimal Dynamic programming algorithm for this problem

On the way to dinner, the CCC competitors are lining up for their delicious curly f

Answer 1

How about divide and conquer? Take a (removable) group somewhere near the middle, and the two groups either side of it, say ...HHHGGGGGHHHHH.... - now there's two possibilities. Either those 2 sets of H's dine in the same group, or they don't. If they dine in the same group, then those G's between them must be removed as a group of precisely those G's (so you may as well try that as your first move). If they don't, then you can solve for the left and right sublists separately. Either case, you've got a shorter list to recurse on.